Method for temporal dispersion correction for seismic simulation, RTM and FWI

ABSTRACT

Method for correcting seismic simulations, RTM, and FWI for temporal dispersion due to temporal finite difference methods in which time derivatives are approximated to a specified order of approximation. Computer-simulated seismic data (51) are transformed from time domain to frequency domain (52), and then resampled using a mapping relationship that maps, in the frequency domain, to a frequency at which the time derivative exhibits no temporal dispersion (53), or to a frequency at which the time derivative exhibits a specified different order of temporal dispersion. Alternatively, measured seismic data from a field survey (61) may have temporal dispersion of a given order introduced, by a similar technique, to match the order of approximation used to generate simulated data which are to be compared to the measured data.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Patent Application 62/009,593, filed Jun. 9, 2014, entitled A Method for Temporal Dispersion Correction for Seismic Simulation, RTM and FWI, the entirety of which is incorporated by reference herein.

FIELD OF THE INVENTION

The invention relates generally to the field of geophysical prospecting for hydrocarbons and, more particularly, to seismic data processing. Specifically, the invention relates to the technical fields of seismic simulation, reverse time depth migration, and full waveform inversion of seismic data to infer a subsurface model of velocity or other physical property. In addition, many other useful industrial simulators for doing electromagnetic propagation, reservoir simulation and heat flow may match the form needed for this invention to apply. As will be explained later in the invention disclosure, the necessary form corresponds to that of a stationary differential operator equation including mixed or non-mixed terms of spatial and temporal derivatives with coefficients that may vary with space but do not vary in time.

BACKGROUND OF THE INVENTION

A seismic simulator used to compute either a forward simulation of a source or the adjoint simulation of a recorded wave field is typically implemented using a time stepping algorithm based upon a selected finite difference approximation to either a first or a second time derivative. Most historical implementations have been for 2nd order time stepping (error is proportional to (Δt)², where Δt is the time step) because that is easy and efficient to implement and requires fewer resources. Using 2nd order time stepping provides a result with temporal dispersion artifacts. Higher-order approximations are better, because the error for approximation of order n is proportional to (Δt)^(n) which→0 as n→∞ for Δt<1. But any finite-order approximation suffers from some degree of temporal dispersion. If the temporal dispersion is not corrected, the application of this type of simulator for forward simulation or to compute Reverse Time Depth Migration (RTM) images and Full Waveform Inversion (FWI) gradients and Hessians will have errors that degrade the value of these products for petroleum exploration and geophysical prospecting.

SUMMARY OF THE INVENTION

In one embodiment, the invention is a method for prospecting for hydrocarbons. Measured seismic data are obtained. Corresponding simulated seismic data are computer-generated using a finite-difference, time-stepping algorithm that approximates a time derivative operator to a selected order of approximation. The simulated seismic data are used to perform full-waveform inversion or reverse-time migration of the measured seismic data, wherein temporal numerical dispersion corresponding to the selected order of approximation is (i) removed from the simulated seismic data or (ii) introduced into the measured seismic data by steps including performing a Fourier transform in time on (i) the simulated or (ii) the measured seismic data, then resampling the transformed seismic data in frequency domain, and then performing an inverse Fourier transform from frequency domain back to time domain. The resampling utilizes a property of a class of stationary finite-difference operators whereby, in frequency domain, an aspect of the temporal numerical dispersion is that a desired numerical solution for a given frequency is computed at an incorrect frequency, and the resampling uses a mapping relationship that maps the incorrect frequency to the given frequency. The full-waveform-inverted seismic data or the reverse-time-migrated seismic data may then be used in known methods to prospect for hydrocarbons. The resampled frequency-domain seismic data may be scaled with a frequency-dependent scaling factor before performing the inverse Fourier transform back to time domain.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention and its advantages will be better understood by referring to the following detailed description and the attached drawings, in which:

FIG. 1 shows a simple two-dimensional earth model for a homogeneous half space with a free surface boundary on top;

FIG. 2 shows simulated seismic data generated using the earth model of FIG. 1 and using a finite difference algorithm that is 2^(nd) order in time and 14^(th) order in space;

FIG. 3 shows simulated seismic data for the same earth model in FIG. 1 but with a much smaller time step for the finite difference in time than the simulation in FIG. 2;

FIG. 4 shows the simulated data from FIG. 2 after applying the present inventive method to correct for temporal numerical dispersion;

FIG. 5 is a flow chart giving basic steps in an embodiment of the present invention that removes temporal numerical dispersion from simulated seismic data by resampling and scaling the transform of the simulated data in the frequency domain;

FIG. 6 is a flow chart giving basic steps in an embodiment of the present invention that applies temporal numerical dispersion to field seismic data, by resampling and scaling the transform of the seismic data in the frequency domain, so that the field data match the simulator data used in RTM or FWI gradient computations;

FIG. 7 shows an example, taken from Stork (2013), of how applying numerical temporal dispersion to the recorded seismic data improves the RTM image when the RTM image is computed using very long spatial derivative operators with an order of accuracy in the range from 24 to 30;

FIG. 8 shows the effects of temporal numerical dispersion on a broadband wavelet after wave propagation, causing the high-frequencies to propagate faster with a larger amount of dispersion visible at later times, which temporal dispersion can be removed by a correction operator;

FIG. 9 is a plot of relative phase speed versus normalized frequency for time differencing for the second order approximation for the temporal second derivative (Tong Fei, 1994);

FIG. 10 is a plot of relative phase speed versus normalized wavenumber for spatial differencing for the second order approximation for the spatial second derivative, showing that the dispersion due to the spatial operator approximately cancels the dispersion due to the temporal operator when the operators are of the same order of accuracy (Tong Fei, 1994);

FIGS. 11A-11E show results of a test example showing the temporal dispersion effects on the seismic wavelet due to finite difference simulation, then applying the present inventive method to remove or add temporal dispersion effects;

FIGS. 12A-12D show an example of how temporal dispersion corrections made with the present inventive method improve the broadband FWI results; and

FIGS. 13A-13C show another example of how temporal dispersion corrections made with the present inventive method improve the broadband FWI results.

The invention will be described in connection with example embodiments. However, to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention, this is intended to be illustrative only, and is not to be construed as limiting the scope of the invention. On the contrary, it is intended to cover all alternatives, modifications and equivalents that may be included within the scope of the invention, as defined by the appended claims.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

To illustrate the problem that the present invention solves, consider an earth model consisting of a simple two-dimensional half-space with a free surface boundary on top as shown in FIG. 1. The p-wave velocity is 2000 m/s. The source is located at x=800 m and z=800 m. The receivers are located on a line at a depth of 1000 m. The only reflector in the model is the free surface boundary. An impulsive source is simulated at zero time and the simulated receiver data due to a finite difference algorithm is shown in FIG. 2. The acoustic finite difference algorithm applied to the model is second order in time and 14^(th) order in space. The vertical axis is time in seconds. The horizontal axis is trace number and there is a 10 m increment between trace locations. A large time step is used in the simulation and the events are distorted by temporal numerical dispersion. High-frequency energy arrives early. Very low-frequency energy arrives at roughly the correct time. The different wave propagation speeds for different frequencies is called dispersion. The exact acoustic wave equation in this case should provide a solution where all frequency components propagate at the same speed. The numerical solution has dispersion because the temporal finite difference operator is inexact. The temporal numerical dispersion is a larger effect for higher frequencies and later times.

FIG. 3 shows simulated seismic data for the same earth model in FIG. 1, but with a much smaller time step for the finite difference in time than the simulation in FIG. 2. These simulated data have less temporal numerical dispersion than shown in FIG. 2 but cost 1.5 times as much to simulate. This is the conventional way of minimizing temporal dispersion. FIG. 4 shows the simulated data from FIG. 2 after applying the present invention to correct for temporal numerical dispersion. The seismic events in FIG. 4 show much more accurate simulations compared to FIG. 2, and compare favorably to the results in FIG. 3. The cost associated with making this correction is very small compared to the cost of the simulation so the cost is almost the same as that for the data in FIG. 2. The invention enables accurate seismic simulation with a large time-step increment and lower cost to match or improve upon the accuracy of a costly seismic simulation with a very small time-step increment.

As stated above, a finite difference operator used for computing a temporal derivative has approximations that create numerical errors when used to solve partial differential equations. It is a realization of the present invention that for certain types of stationary differential operators that incorporate approximate temporal derivative operators, the correct solution is computed by the approximate equation, but at the wrong frequency. The invention uses this feature to correct an approximate solution into a more correct solution by a resampling operation in the frequency domain.

Virtually all of the time-domain forward and adjoint wave simulation algorithms used for seismic simulation, seismic migration and seismic full waveform inversion correspond to the type of stationary differential operators to which this invention applies. Electromagnetic equations and heat flow problems can also be formulated in a way such that the theory developed here applies. Thus, a wide range of time simulation processes may be able to use this method.

The form necessary for the invention to apply corresponds to that of a stationary differential operator equation including mixed or non-mixed terms of spatial and temporal derivatives with coefficients that may vary with space but do not vary in time. The differential operators for time stepping can be either explicit or implicit. For seismic simulation, RTM, and FWI applications, the basic import of this is that the earth model properties do not change during one seismic simulation. Alternatively, this assumption would be violated if the earth model were to change during the simulation. One example of a violation of this assumption would be if moving water waves on the air-water boundary were to make the earth model properties change with time during the simulation.

The present invention differs from Stork (2013) in many ways: (a) how the temporal numerical dispersion is corrected or applied (The invention uses Fourier-domain resampling instead of Stork's choice of time-domain filter banks to implement temporal numerical dispersion corrections.); (b) by extending the application to FWI objective function, gradient and Hessian computations; (c) by modifying dispersion from approximate derivative operators of any order of accuracy to any other order of accuracy and thereby enabling the match of temporal operator order to spatial operator, which enables (d) the application of temporal dispersion correction operators or their inverse to less perfect spatial operators. Shorter spatial operators enable more efficient halo exchanges for parallel domain-decomposed computations. The combination of all of these aspects can lead to improved efficiencies for a given level of accuracy for simulators, RTM applications and FWI applications.

The present invention differs from the patent application by Zhang et al. (2012) because the key step in the process is resampling in the frequency domain rather than filtering in the frequency domain. Filtering typically implies convolution but this invention is instead based upon resampling to change variables. The invention also has a more general range of applicability to a specific class of stationary differential operator equations. As a result the present invention applies to time-stepping differential equations in other fields, e.g. heat flow and reservoir simulation, in addition to seismic simulation and seismic migration. Zhang does not teach us to apply his method to FWI or other seismic applications.

Computing ω_(approx) (ω) for an Example Finite Difference Operator

A finite difference solution applies approximate derivative operators to solve a problem. The approximate derivative operators in the time domain can be Fourier transformed to the frequency domain and compared to the exact form of the derivative, which is iω.

The Fourier transform F(ω) of an explicit finite difference operator with coefficients f_(j) at times t_(j) is the Fourier transform of a digital filter. It is computed by the following equation.

$\begin{matrix} {{F(\omega)} = {\sum\limits_{j = 0}^{N - 1}\;{{f_{j}\left( t_{j} \right)}e^{{- i}\;\omega\; t_{j}}}}} & (1.1) \end{matrix}$ On a regular grid, a second derivative finite difference operator is a symmetric digital filter.

Consider for example the explicit convolutional centered second derivative in time finite difference operator with n_(order)+1 coefficents. Here, the coefficients a_(j) represent the zero lag and positive lags of a symmetric filter used to approximate the second derivative operator. Those coefficients can be Taylor series coefficients or be optimized coefficients designed to fit a specified bandwidth with high accuracy. The exact operator in the Fourier domain would be −ω². On a regular grid, a second derivative finite difference operator is a symmetric digital filter. The Fourier transform of a symmetric digital filter is a cosine transform

$\begin{matrix} {{- {\omega_{approx}^{2}\left( {n_{order},\omega,{\Delta\; t}} \right)}} = \frac{a_{0} + {2{\sum\limits_{j = 1}^{n_{order}/2}\;{a_{j}{\cos\left( {\omega\; j\;\Delta\; t} \right)}}}}}{\Delta\; t^{2}}} & (1.2) \end{matrix}$ where n_(order) is the order of the finite difference approximation and Δt is the finite time step. Comparing the exact operator in the Fourier domain to the approximate finite difference operator in the Fourier domain leads to the following relationship between approximate angular frequency and true angular frequency.

$\begin{matrix} {{\omega_{approx}\left( {n_{order},\omega,{\Delta\; t}} \right)} = \frac{\sqrt{{- a_{0}} - {2{\sum\limits_{j = 1}^{n_{order}/2}\;{a_{j}{\cos\left( {j\;{\omega\Delta}\; t} \right)}}}}}}{\Delta\; t}} & (1.3) \end{matrix}$ The error for this order of approximation will be proportional to (Δt)^(norder) when equation (1.3) applies to explicit temporal derivative operators derived from Taylor series expansions.

The second order approximation for the angular frequency made while doing a finite difference temporal second derivative corresponds to the following in the frequency domain as a function of the true angular frequency ω and the time step increment Δt.

$\begin{matrix} {{\omega_{approx}\left( {2,\omega,{\Delta\; t}} \right)} = {\frac{\sqrt{2 - {2\;{\cos\left( {{\omega\Delta}\; t} \right)}}}}{\Delta\; t} = {\frac{2}{\Delta\; t}{\sin\left( \frac{{\psi\Delta}\; t}{2} \right)}}}} & (1.4) \end{matrix}$ The inverse mapping from approximate angular frequency to true angular frequency can also be made.

$\begin{matrix} {{\omega\left( {\omega_{approx},{\Delta\; t}} \right)} = {{\frac{1}{\Delta\; t}{\cos^{- 1}\left( {1 - \frac{\left( {\omega_{approx},{\Delta\; t}} \right)^{2}}{2}} \right)}} = {\frac{2}{\Delta\; t}{\sin^{- 1}\left( \frac{\omega_{approx}\Delta\; t}{2} \right)}}}} & (1.5) \end{matrix}$ The examples above have been for explicit convolutional-style temporal derivative operators. This type of mapping can also be done for implicit operators as described by Crank and Nicolson (1947) that could be designed to have properties of unconditional stability for time stepping with large time increments. The most common Crank-Nicolson approach as implemented by Claerbout (1985) for a wave equation would use the second-order bilinear Z transform. For that case, this mapping would apply:

$\begin{matrix} {{\omega_{approx}\left( {\omega,{\Delta\; t}} \right)} = {\frac{2}{\Delta\; t}{{\tan\left( \frac{{\omega\Delta}\; t}{2} \right)}.}}} & (1.6) \end{matrix}$ Comparing Solutions for Stationary Differential Operator Equations

Consider a differential operator L operating on a function u equal to a broad band source term s(x,t). Choose L to be a linear sum of terms, each scaled by the spatially varying coefficients and/or by spatial derivatives to any order and/or by mixed spatial derivatives to any order and/or by time derivatives to any order. The operator varies spatially with the k^(th) operator coefficient term c_(k) optionally a function of position x. However the operator is stationary with respect to time in that operator coefficients c_(k) are not time dependent. The operator L may be dependent upon temporal derivatives of any order but not on time explicitly.

$\begin{matrix} {{{L\left( {{c_{k}(x)},\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial t}} \right)}{u\left( {x,t} \right)}} = {s\left( {x,t} \right)}} & (1.7) \end{matrix}$ If the operator L contains time derivatives of any order but no coefficients that vary with time, then the frequency-domain equivalent operator {tilde over (L)} retains a similar form with each time derivative replaced by iω. Such an operator L is of the type of stationary differential operator to which the present inventive method applies.

Then if U(x,ω) is the temporal Fourier transform of the solution wavefield u(x,t) and S(x,ω) is the temporal Fourier transform of the source term s(x,t),

$\begin{matrix} {{{\overset{\sim}{L}\left( {{c_{k}(x)},\frac{\partial}{\partial x_{i}},\omega} \right)}{U\left( {x,\omega} \right)}} = {{S\left( {x,\omega} \right)}.}} & (1.8) \end{matrix}$ The Helmholtz equation (Morse and Feschbach, 1953) is an example of a differential equation with the form given in equation 1.8.

The following two operator equations have identical solutions for ω₂=ω₁. Mathematically, this is a trivial statement since the two equations are identical except that the variables have been renamed. The key point is that ω₁ and ω₂ can represent different temporal derivative operators. This is how one can recognize that the approximate solution contains within it the true solution at the wrong frequency.

$\begin{matrix} {{{\overset{\sim}{L}\left( {{c_{k}(x)},\frac{\partial}{\partial x_{i}},\omega_{1}} \right)}{U\left( {x,\omega_{1}} \right)}} = {S\left( {x,\omega_{1}} \right)}} & (1.9) \\ {{{\overset{\sim}{L}\left( {{c_{k}(x)},\frac{\partial}{\partial x_{i}},\omega_{2}} \right)}{U\left( {x,\omega_{2}} \right)}} = {S\left( {x,\omega_{2}} \right)}} & (1.10) \end{matrix}$

In practice, the solution U₁(x,ω₁) to the operator ω₂(ω₁) is usually computed for the following equation.

$\begin{matrix} {{{\overset{\sim}{L}\left( {{c_{k}(x)},\frac{\partial}{\partial x_{i}},{\omega_{2}\left( \omega_{1} \right)}} \right)}{U_{1}\left( {x,\omega_{1}} \right)}} = {S\left( {x,\omega_{1}} \right)}} & (1.11) \end{matrix}$ If ω₁=ω_(exact) and ω₂(ω₁) is the Fourier transform of a temporal finite difference operator, then U₁(x,ω₁) would represent the solution to a specific temporal finite difference approximation. That solution can be mapped by a resampling and scaling operator in the frequency domain into a solution U₂(x,ω₂) with exact temporal derivatives consistent with equation (1.12) as follows.

$\begin{matrix} {{U_{2}\left( {x,\omega_{2}} \right)} = {{U_{1}\left( {x,{\omega_{2}\left( \omega_{1} \right)}} \right)}\frac{S\left( {x,{\omega_{2}\left( \omega_{1} \right)}} \right)}{S\left( {x,\omega_{1}} \right)}}} & (1.12) \end{matrix}$ Basic steps in this embodiment of the present inventive method are shown in the flow chart in FIG. 5. Note that U₁(x,ω₁) and U₂(x,ω₂) could be solutions to different operator equations if the meaning of ω₁ and ω₂ are different in the context of approximate and exact temporal derivative operators. Then the same mapping given in equation (1.12) still applies but in a different context. Basic steps in a second embodiment of the present inventive method suitable to be used for RTM and FWI applications are given in the flow chart in FIG. 6.

Equation 1.12 is quite general. However, one limitation is that if multiple seismic sources are being simultaneously simulated in a single simulation, as in simultaneous-source FWI or simultaneous source RTM (see, for example, U.S. Pat. No. 8,121,823 to Krebs, et al.), then the simultaneous sources need to have the same source time functions to within a scale factor. They can vary by scale factors of +1 and −1. They may be in multiple spatial locations.

In summary, the invention applies to correcting or modifying temporal numerical dispersion characteristics associated with solutions to stationary differential operator equations of the style discussed above. In one of its FIG. 5 embodiments, the present inventive method comprises the following basic steps:

-   (1) Begin with a solution u₁(x,t₁) to a stationary differential     operator equation Lu₁(x,t₁)=s(x,t₁) using a known approximate or     true temporal derivative operator. This solution is the simulated     seismic data 51 in the flow chart of FIG. 5. -   (2) In step 52, do a temporal Fourier transform from time t₁ to     angular frequency ω₁ converting u₁(x,t₁) into U₁(x,ω₁).

$\begin{matrix} {{U_{1}\left( {x,\omega_{1}} \right)} = {\int_{- \infty}^{\infty}{{u_{1}\left( {x,t_{1}} \right)}e^{{- i}\;\omega_{1}t_{1}}\ d\; t_{1}}}} & (1.13) \end{matrix}$

-   (3) Take ω₁(ω) to be a bijective function over some range of ω that     relates the approximate or true temporal derivative operator     associated with ω₁ used for solution U₁(x,ω₁) to the true ω used for     the exact solution U(x,ω). This can be readily computed by taking a     Fourier transform of the approximate operator and comparing that to     the Fourier transform of an exact operator as discussed in a prior     section. Likewise, take ω₂(ω) to be a bijective function over some     range of ω that relates the approximate or true temporal derivative     operator ω₂ used for solution U₂(x,ω₂) to the true ω used for the     exact solution U(x,ω). The invention converts an available solution     U₁(x,ω₁) with the temporal numerical dispersion characteristics of     temporal derivative operators associated with ω₁(ω) to match a     desired solution U₂(x,ω₂) with the temporal numerical dispersion     characteristics of temporal derivative operators associated with     ω₂(ω) by finding the bijective relationship ω₂(ω₁) and resampling     U₁(x,ω₁) to get the desired values of U₂(x,ω₂) via equation (1.12).     This is step 53. -   (4) In step 54, perform an inverse temporal Fourier transform of     U₂(x,ω₂) to get the temporal dispersion corrected or modified result     u₂(x,t₂) in the time domain (55).

$\begin{matrix} {{u_{2}\left( {x,t_{2}} \right)} = {\frac{1}{2\pi}\;{\int_{- \infty}^{\infty}{{U_{2}\left( {x,\omega_{2}} \right)}e^{i\;\omega_{2}t_{2}}\ d\;\omega_{2}}}}} & (1.14) \end{matrix}$

Next, some aspects of the invention are described in more detail. The embodiment of the present invention implementing temporal numerical dispersion corrections, via resampling in the temporal Fourier domain, to simulated seismic data can be applied as a post-processing single-seismic-trace-at-a-time process applied either within the seismic simulator or as a separate seismic processing application. A flow chart showing basic steps in this embodiment of the present inventive method is given in FIG. 5. The associated computation time is negligible compared to the computation time needed for the simulator. The simulation time step and the source excitation time are required parameters and therefore incorporating this correction within the simulator application can be very convenient. If the application is done by a separate application, the simulation time step must be known in the separate application.

For a second embodiment of the present invention that applies temporal numerical dispersion via resampling in the temporal Fourier domain to field data to be input to RTM or FWI gradient or Hessian computation, this may be done as a preprocessing step. Basic steps in this embodiment are shown in FIG. 6, and it can be seen that the steps are the same as in FIG. 5, except that they are applied to the measured data 61 instead of the simulated data. However a different mapping relationship (not shown in the flow charts) will be used in the resampling step in the embodiment of FIG. 6, mapping frequency to a different ω₂ than would typically be used in the embodiment of FIG. 5. Because the time step of the simulator used internally in the RTM or FWI application is required to be known, this step is most conveniently done as part of the RTM or FWI application, but can be done as separate pre-processing single-seismic-trace-at-a-time filter application. FIG. 7 taken from Stork (2013) shows the impact on an RTM image of applying temporal numerical dispersion to the input field data. It should be noted that Stork does this by method that is different from the present invention.

It may be noted that full waveform inversion can be corrected for temporal dispersion using either embodiment of the present inventive method, i.e. that of FIG. 5 or FIG. 6. In other words, the simulated data can be corrected to remove dispersion, or dispersion can be applied to the measured data to match the degree of dispersion introduced by whatever order of approximation the simulation time-stepping algorithm used to compute the time derivatives. As long as the simulated data and measured data have the same order of dispersion, the dispersion will cancel out when the objective function is computed to use for the model update.

The present invention can be used to either apply or remove all temporal numerical dispersion as shown in FIG. 8. The input has a broadband wavelet. After wave propagation, temporal numerical dispersion causes the high-frequencies to propagate faster with a larger amount of dispersion visible at later times. The temporal dispersion can be removed by a correction operator.

The present invention can be applied to change the temporal numerical dispersion characteristics of field or synthetic seismic data from any operator order to any other operator order. Equation 1.3 gives the relationship between true frequency and approximate frequency for explicit centered temporal finite difference operators on a regular grid. This relationship is objective over the specific range of frequency and time increment parameters of interest. Therefore the inverse relationship can be found. Therefore one can map any operator to the true operator and then back to another approximate operator. A general expression for the inverse may be difficult to write for some operator choices, but a computer can easily tabulate these and then look up values in the table to solve the inverse relationship.

The effect of a second order temporal finite difference operator on phase velocity has been described by Fei (1994) and is shown in FIG. 9. The compensating errors for the same order of spatial operator are shown by Fei (1994) in FIG. 10. The error compensation is exact in one dimensional wave propagation but only approximate for two-dimensional and three-dimensional wave propagation. Accuracy is often improved by matching time derivative operators to the same order of accuracy as chosen for the spatial derivative operators. The logic behind making this choice is discussed by Aldridge and Haney (2008) where they recommend always matching the order of approximation for time derivatives and spatial derivatives. This choice can be made for either forward simulation or for RTM and FWI gradient and Hessian applications.

A key aspect of the methods disclosed herein, as indicated by equation (1.12), is that the seismic data can be advantageously modified from one form of ω operator to another form by resampling in the frequency domain. The input and output ω operators can be exact or approximate, and if approximate, they can be explicit or implicit.

A test of the present invention is shown in FIGS. 11A-11E, mapping temporal numerical dispersion from one order of accuracy to another. FIG. 11A shows three input spikes at periodic intervals for a finite difference computer simulation. FIG. 11B shows what they look like due to temporal numerical dispersion after time steps with the time derivative approximated at order 2. The dispersion was added using the FIG. 6 embodiment of the present invention. FIG. 11C shows how the result of FIG. 11B is corrected to change the temporal numerical dispersion from order 2 to order 8 using the present inventive method. This might be done to match eighth-order spatial operators for a code that had spatial numerical dispersion. FIG. 11D shows the result after mapping of 11C back to 2^(nd) order. The almost exact duplication of FIG. 11B shows how invertible the present inventive method is. To generate FIG. 11E, the present invention is applied to correct dispersion from order 2 to exact, i.e. the result in 11B (or 11D) is mapped to exact—all temporal numerical dispersion is removed.

An ideal simulation result with no temporal dispersion would be three spikes, i.e. this would be the ideal result for FIG. 11B. However, a simulator with time derivative operators that are second order accurate is used and creates data with temporal dispersion high frequency components that arrive early and the waveform shape varies with time. The dispersive simulated data with second order accuracy in time are converted into data with eighth order accuracy in time. These data can be mapped back to second order accuracy in time or back to data with no temporal numerical dispersion.

Low-frequency FWI is less affected by temporal numerical dispersion than broad-band FWI because temporal numerical dispersion is less important at low frequencies. High resolution images of the subsurface require broad band FWI, and temporal numerical dispersion corrections become more important. Temporal numerical dispersion corrections become very important when making accurate ties between inverted FWI earth model parameters and well logs.

FIGS. 12A-12D provide an example of how temporal dispersion corrections improve broadband FWI results. The initial model input to a first iteration of FWI, shown in FIG. 12A, is a very smooth representation of the subsurface and lacks detail. The true layered model for this simulation and inversion is given in FIG. 12B. The estimated FWI model doing FWI without including temporal numerical dispersion corrections is given in FIG. 12C. It may be noted how the layer boundaries are fuzzy and that some high-wavenumber events appear within the blocky layers. When temporal numerical dispersion corrections (made by the present inventive method) are included in the FWI earth model estimation process, the result, shown in FIG. 12D, is much closer to the true model.

FIGS. 13A-13C provide an example of how temporal dispersion corrections improve the broadband FWI results for another earth model, one with a more realistic “true” velocity model. Comparisons are made between synthetic well data (gray) versus the earth model. In FIG. 13A, the smooth initial earth model is compared to the true earth model. In FIG. 13B, the inverted FWI model without dispersion corrections is compared to the true earth model. In FIG. 13C, the FWI model with temporal numerical dispersion corrections are compared with the true earth model. Note that the improved result in 13C was generated by first applying temporal numerical dispersion to the true data, using the embodiment of the invention illustrated in FIG. 6, prior to running FWI with pre-defined simulation time-step of 0.001 sec and spatial finite difference order fourteen. Inaccuracies in the phase of the inverted model shown in FIG. 13B relative to the true model have been corrected using the invention. Computational costs for both 13B and 13C are similar. A geologist with expertise in well-log interpretation would note that the inverted FWI result with temporal numerical dispersion correction included would provide a significantly different interpretation of potential reservoir properties than the FWI result without the correction.

The foregoing description is directed to particular embodiments of the present invention for the purpose of illustrating it. It will be apparent, however, to one skilled in the art, that many modifications and variations to the embodiments described herein are possible. All such modifications and variations are intended to be within the scope of the present invention, as defined by the appended claims.

REFERENCES

-   Aldridge, David F., and Matthew M. Haney, “Numerical dispersion for     the conventional-staggered-grid finite-difference elastic wave     propagation algorithm,” Sandia National Laboratories, SAND2008-4991     (2008). -   Claerbout, J., Imaging the Earth's Interior, Blackwell Scientific     Publications, 96-99, 104, 116, 126, 141, 256-257, 265, 284, 305-307     (1985); this book may be viewed online at     Stanford.edu/data/media/public/sep//prof/index. -   Crank, J., and P. Nicolson, “A practical method for numerical     evaluation of solutions of partial differential equations of the     heat conduction type”. Proc. Camb. Phil. Soc. 43 (1), 50-67(1947). -   Fei, Tong, “Elimination of numerical dispersion in finite difference     modeling and migration by flux-corrected transport,” Ph.D. thesis,     Colorado School of Mines (1994), particularly FIGS. 3.1 and 3.2;     cwp.mines/edu/researchpublications/CWPresearchreports. -   Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, Part     I., New York, McGraw-Hill, pp. 125-126, 271, and 509-510 (1953). -   Stork, Christof, “Eliminating nearly all dispersion error from FD     modeling and RTM with minimal cost increase,” EAGE expanded     abstracts (2013). -   Zhang, Linbin, Guojian Shan, Yue Wang, United States Patent     Application Publication US 2012/0243371 A1(Sep. 27, 2012). 

The invention claimed is:
 1. A method for prospecting for hydrocarbons, comprising: obtaining measured seismic data; generating, with a computer, simulated seismic data using a finite-difference, time-stepping algorithm that approximates a time derivative operator to a selected order of approximation; performing, with the computer, full waveform inversion or reverse time migration of the measured seismic data with the simulated seismic data, wherein temporal numerical dispersion corresponding to the selected order of approximation is (i) removed from the simulated seismic data or (ii) introduced into the measured seismic data by steps including, performing, with the computer, a Fourier transform in time on (i) the simulated or (ii) the measured seismic data, then resampling the transformed seismic data in frequency domain, and then performing an inverse Fourier transform from frequency domain back to time domain, wherein said resampling utilizes a property of a class of stationary finite-difference operators, wherein in frequency domain, an aspect of the temporal numerical dispersion is that a desired numerical solution for a given frequency is computed at an incorrect frequency, and said resampling uses a mapping relationship that maps the incorrect frequency to the given frequency; and prospecting for hydrocarbons with the full-waveform-inverted seismic data or the reverse-time-migrated seismic data.
 2. The method of claim 1, further comprising scaling the resampled frequency-domain seismic data with a frequency-dependent scaling factor before performing the inverse Fourier transform back to time domain.
 3. The method of claim 2, wherein simulation of the seismic data comprises a wave propagation equation with a stationary, finite-difference differential operator and a source term S in frequency domain, and wherein the scaling factor can be expressed as $\frac{S\left( {x,{{\omega 2}({\omega 1})}} \right)}{S\left( {x,{\omega 1}} \right)}$ where ω₁ is the incorrect frequency, ω₂ is the given frequency, and x is spatial location of the source.
 4. The method of claim 1, wherein the time derivative being approximated by the finite-difference algorithm is a centered second derivative, the given frequency is true frequency, and the mapping relationship can be expressed as ${\omega_{approx}\left( {n_{order},\omega,{\Delta\; t}} \right)} = \frac{\sqrt{{- a_{0}} - {2{\sum\limits_{j = 1}^{n_{order}/2}\;{a_{j}{\cos\left( {j\;{\omega\Delta}\; t} \right)}}}}}}{\Delta\; t}$ where ω_(approx) is the incorrect frequency, ω is the true frequency, n_(order) is the selected order of approximation, Δt is duration of a time step in the algorithm, a_(j) are coefficients of a symmetric filter used to approximate the second derivative operator, and j is an integer index ranging from 1 to n_(order)/2.
 5. The method of claim 1, wherein said class of stationary finite-difference operators includes any differential operator that operates on a function of spatial position and time to equate to a source term, where the differential operator includes at least one spatial derivative of any order, at least one time derivative of any order, and may vary with position but is constant with time.
 6. The method of claim 1, wherein temporal numerical dispersion is removed from simulated seismic data by using a mapping relationship that maps the incorrect frequency to a true frequency, being a frequency at which the simulation generates a solution for the incorrect frequency with no temporal numerical dispersion.
 7. The method of claim 6, wherein spatial derivatives in the finite difference time stepping algorithm are approximated to order at least
 20. 8. The method of claim 1, wherein temporal numerical dispersion is removed from simulated seismic data by using a mapping relationship that maps the incorrect frequency to a frequency at which the simulation generates a solution having temporal numerical dispersion of a same order of approximation as a spatial derivative approximation in the finite-difference algorithm.
 9. The method of claim 1, wherein temporal numerical dispersion corresponding to the selected order of approximation in the algorithm is introduced into the measured seismic data to match the temporal numerical dispersion present in the simulated seismic data, said introduction of temporal numerical dispersion into the measured seismic data using a mapping relationship that is an inverse of a frequency mapping relationship that would remove all temporal numerical dispersion from the simulated seismic data.
 10. The method of claim 1, wherein prospecting for hydrocarbons comprises causing a well to be drilled at a location identified using the full-waveform-inverted seismic data or the reverse-time-migrated seismic data.
 11. A method for prospecting for hydrocarbons, comprising: obtaining measured seismic data; simulating, with a computer, seismic data to correspond to the measured seismic data using a finite-difference, time-stepping algorithm programmed on a computer, which algorithm approximates a time derivative operator to a selected order of approximation; removing, with the computer, temporal numerical dispersion caused by the approximation from the simulated seismic data using steps including, Fourier transforming, with the computer, the simulated seismic data to frequency domain, wherein a time variable is transformed to a frequency variable, resampling, with the computer, the simulated seismic data in the frequency domain, and inverse-transforming, with the computer, the resampled simulated seismic data back to time domain; reverse-time migrating, with the computer, the measured seismic data with the resampled data in time domain, or inverting, with the computer, full-wavefield inversion the measured seismic data with the resampled data in time domain; generating a subsurface image from the reverse-time migration or a subsurface model from the full-wavefield inversion; and prospecting for hydrocarbons with the subsurface image or the subsurface model.
 12. The method of claim 11, wherein the resampling maps the simulated seismic data for a given frequency to a different frequency at which the temporal numerical dispersion is removed.
 13. The method of claim 11, wherein prospecting for hydrocarbons comprises causing a well to be drilled at a location identified using the subsurface image or the subsurface model.
 14. A method for prospecting for hydrocarbons, comprising: obtaining measured seismic data; simulating, with a computer, seismic data to correspond to the measured seismic data using a finite-difference, time-stepping algorithm programmed on a computer, which algorithm approximates a time derivative operator to a selected order of approximation; introducing, with the computer, temporal numerical dispersion into the measured seismic data to match temporal numerical dispersion caused in the simulated seismic data by the selected order of approximation, using steps comprising: Fourier transforming, with the computer, the measured seismic data to frequency domain, wherein a time variable is transformed to a frequency variable; resampling, with the computer, the measured seismic data in the frequency domain; inverse-transforming, with the computer, the resampled measured seismic data back to time domain; reverse-time migrating or full-wavefield inverting the time domain resampled measured seismic data with the simulated seismic data; generating a subsurface image from the reverse-time migration or a subsurface model from the full-wavefield inversion; and prospecting for hydrocarbons with the subsurface image or the subsurface model.
 15. The method of claim 14, wherein the resampling maps the measured seismic data for a given frequency to a different frequency at which temporal numerical dispersion corresponding to the order of approximation in the algorithm is introduced into the measured seismic data.
 16. The method of claim 14, wherein prospecting for hydrocarbons comprises causing a well to be drilled at a location identified using the subsurface image or the subsurface model. 